The detection variant was then addressed by Censor-Hillel, Kaski, Korhonen, Lenzen, Paz, and Suomela [1]}, who show a deterministic constant-round algorithm for detecting 4-cycles. In addition, they show an \(O(n^{0.158})\) -round algorithm for detecting \(p\) -cycles, for any constant \(p\) . More accurately, the complexity is \(2^{O(p)}n^{0.158}\)  rounds. This is a randomized algorithm that relies on the color coding technique of Alon, Yuster, and Zwick [2]}, in order to avoid too much congestion that would be caused by collecting all possible candidates for cycle nodes. More recently, Censor-Hillel, Fischer, Gonen, Le Gall, Leitersdorf, and Oshman [3]} showed that \(2p\) -cycles can be detected in \(O(1)\)  rounds for any \(p\) , using a technique which also allows fast girth approximation. This is, notably, a deterministic algorithm. For odd values of \(p\) , it is still not known whether there is a constant-round \(p\) -cycle detection algorithm.
